Classical dynamics and localization of resonances in the high-energy ...
PHYSICAL REVIEW E 91, 012915 (2015)
Classical dynamics and localization of resonances in the high-energy region of the hydrogen atom in crossed fields Frank Schweiner, J¨org Main, Holger Cartarius, and G¨unter Wunner Institut f¨ur Theoretische Physik 1, Universit¨at Stuttgart, 70550 Stuttgart, Germany (Received 7 October 2014; published 20 January 2015) When superimposing the potentials of external fields on the Coulomb potential of the hydrogen atom, a saddle point (called the Stark saddle point) appears. For energies slightly above the saddle point energy, one can find classical orbits that are located in the vicinity of this point. We follow those so-called quasi-Penning orbits to high energies and field strengths, observing structural changes and uncovering their bifurcation behavior. By plotting the stability behavior of those orbits against energy and field strength, the appearance of a stability apex is reported. A cusp bifurcation, located in the vicinity of the apex, will be investigated in detail. In this cusp bifurcation, another orbit of similar shape is found. This orbit becomes completely stable in the observed region of positive energy, i.e., in a region of parameter space, where the Kepler-like orbits located around the nucleus are already unstable. By quantum mechanically exact calculations, we prove the existence of signatures in quantum spectra belonging to those orbits. Husimi distributions are used to compare quantum-Poincar´e sections with the extension of the classical torus structure around the orbits. Since periodic orbit theory predicts that each classical periodic orbit contributes an oscillating term to photoabsorption spectra, we finally give an estimation for future experiments, which could verify the existence of the stable orbits. DOI: 10.1103/PhysRevE.91.012915
PACS number(s): 32.60.+i, 32.80.−t, 05.45.−a
I. INTRODUCTION
The hydrogen atom in crossed electric and magnetic fields is a simple example of a nonintegrable physical system, which has been investigated for almost 100 years, theoretically [1–5] as well as experimentally [3–14]. Furthermore, it has been used for studies on phenomena such as Ericson fluctuations [8,15] or Arnol’d diffusion [6]. Even quantum dots [16] and excitons [17] in condensed-matter physics can be explored using the findings attained from the hydrogen atom in crossed fields. One of the major purposes in recent decades was to uncover how chaos, which can be observed in a classical treatment [18,19], shows itself in quantum spectra, since the Schr¨odinger equation is, due to its linearity, not capable of producing chaotic behavior [5,20–26]. Nevertheless, new phenomena occur in crossed fields: the so called quasiPenning resonances [27]. From a classical point of view, those resonances describe a movement of the electron that is localized around a saddle point in the potential, namely the Stark saddle point. The possible appearance of wave functions localized far away from the nucleus led to a series of further investigations [27–32], since they can play an important role in the ionization process. Transition state theory predicts classical orbits localized around the Stark saddle point and states that all ionizing orbits have to pass in the vicinity of this point [33–36]. Even though Clark et al. [27] already proved their existence in 1985, it was not until 2009 that the first signatures of those orbits were found in calculated quantum spectra [37]. The classical stability behavior of the quasi-Penning orbits at high energies and field strengths was first investigated by Fl¨othmann in 1994 [38]. He uncovered a stability apex in parameter space if the stability change of the orbits is illustrated as a function of both parameters. It is the purpose of this paper to perform more precise calculations on the stability of those orbits and to uncover the bifurcation behavior as well as the processes taking place around the stability apex. We report the existence of a cusp 1539-3755/2015/91(1)/012915(11)
bifurcation, which appears in the vicinity of the stability apex and involves another orbit of similar shape. This orbit, becoming completely stable in a specific area of parameter space, is the starting point for semiclassical and quantummechanical calculations. We will demonstrate that signatures of those orbits at high energies and field strengths can be found in accurately calculated quantum spectra. The paper is organized as follows. In Sec. II, the system is introduced, and a scaling of parameters as well as a regularization of coordinates are carried out. A comparison between exceptional points and a classical cusp bifurcation is drawn. The classical stability of the quasi-Penning resonances and their bifurcation behavior at high energies and field strengths are presented in Sec. III. An introduction into the semiclassical and exact quantum-mechanical calculations performed is given in Sec. IV, before pertinent results are discussed in Sec. V. In Sec. VI, a short summary is given and conclusions are drawn. II. CLASSICAL CALCULATIONS A. Hamiltonian, monodromy matrix, and regularized coordinates
The Hamiltonian of a hydrogen atom in a constant electric field F = F eˆx and a constant magnetic field B = B eˆz reads in atomic units (Hartree units with lengths, energies, and electric and magnetic field strengths given in units of 5.29 × 10−11 m, 4.36 × 10−18 J, 5.14 × 1011 V/m, and 2.35 × 105 T, respectively) H =
1 2 1 1 1 p − + BLz + B 2 (x 2 + y 2 ) + F x, 2 r 2 8
(1)
with Lz = xpy − ypx . In the following, we shall take advantage of a scaling property of the classical Hamiltonian enabling us to deal with only two independent variables, i.e., the scaled
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energy and the scaled field strength. The other variables have to be scaled as well, so that the scaling transformations read E˜ = EB −2/3 , F˜ = F B −4/3 , r˜ = rB 2/3 , p˜ = pB −1/3 , t˜ = tB.
(4)
1 L(U) P, r together with a transformation of time, dt = 2r dτ.
(6b)
(6c)
The equations of motion (3) and (5) remain the same except for the replacements γ → = (U, P)T and t → τ . The Hamiltonian then reads H = 12 P 2 − [E − F (U1 U3 − U2 U4 )]U 2
+ 12 (U1 P2 − U2 P1 ) U32 + U42 + (U3 P4 − U4 P3 ) U12 + U22 (7) + 18 U 2 U12 + U22 U32 + U42 = 2, with U 2 = 4i=1 Ui2 . By integrating two further equations along with the Hamiltonian equations of motion, one obtains
(8)
(9)
The transformation (6) is not bijective. For this reason, the inversion, ⎞ ⎛√ r + z cos ϕ+α 2 ⎜ √r + z sin ϕ+α ⎟ ⎜ 2 ⎟ (10) U = ⎜√ ⎟, ⎝ r − z cos ϕ−α ⎠ 2 √ r − z sin ϕ−α 2 contains an additional parameter α, which, without loss of generality, we will set to the constant value α = 0. B. Cusp bifurcation and exceptional points
A cusp bifurcation, appearing in systems with at least two parameters a and b, is described by the normal form [42] x˙ =
(5)
¯ The eigenvalues of M(t) therefore indicate whether a variation of the initial conditions of a periodic orbit leads to exponential divergence or a trajectory remains in the vicinity of the periodic orbit for all times. In the case of a Hamiltonian system, the energy conservation leads to two variational directions not affecting the system’s behavior. The corresponding eigenvalues hence take on the value of 1. Omitting those directions, one obtains the so-called monodromy matrix M. To prevent a divergence of the momentum p near the nucleus, the Kustaanheimo-Stiefel regularization of coordinates [40,41] is used, ⎛ ⎞ U3 −U4 U1 −U2 1 ⎜U U3 U2 U1 ⎟ r = L(U)U = ⎝ 4 U, (6a) U2 −U3 −U4 ⎠ 2 U1 U2 −U1 −U4 U3 p=
d 1
S = P 2 + (U1 P2 − U2 P1 ) U32 + U42 dτ 2 + (U3 P4 − U4 P3 ) U12 + U22 .
(2b)
The Stark saddle point is characterized by the √ fixed-point condition γ˙ = 0, yielding its position r = (−1/ F , 0, 0)T SP √ and energy ESP = −2 F . The stability of orbits is investi¯ M ¯ describes in a linear gated using the stability matrix M. approximation the relative behavior of two trajectories γ (1) and γ (2) in time [39],
and it can be determined by its equation of motion ∂ 2H ¯ d ¯ 0 1 ¯ M= J M with J = , M(0) = 1. −1 0 dt ∂γ 2
d t = U 2, dτ
(2a)
Without further usage of the tilde sign, the Hamiltonian has the same structure as in Eq. (1) when setting B = 1. Defining γ = (r, p)T , we want to find periodic solutions of the Hamiltonian equations of motion, ∂H d 0 1 γ = J with J = . (3) −1 0 dt ∂γ
(1) ¯ γ (1) (t) − γ (2) (t) = M(t)[γ (0) − γ (2) (0)],
the periods and actions of the orbits:
4 3 x 27
+ ax + b.
(11)
4 We choose the factor in front of x 3 as 27 instead of 1 to simplify the results. When calculating the fixed points of Eq. (11), one finds three real solutions for a < 0 and |b| < (−a)3/2 . The two boundary lines of this area are tangent bifurcation lines, along which two fixed points coincide. In the remaining parameter space, only one real solution can be found. A specific attribute of the cusp bifurcation shows a similarity to exceptional points: Following one fixed point around the cusp point, which is located at (a, b) = (0, 0), along a circle,
(a, b) = (−r cos ϕ, − r sin ϕ) with ϕ ∈ [0, 2π ),
(12)
it can be transformed into another one. This phenomenon appears in the case of exceptional points, too. An exceptional point, first described by Kato [43], marks the coalescence of at least two resonances (or more precisely, eigenvalues and corresponding eigenvectors) of a complex Hamiltonian in an at least two-dimensional parameter space [44–47]. Encircling the exceptional point in parameter space, the resonances permute. In the case of an EP2, two resonances interchange and the initial situation can be restored after two cycles [Figs. 1(a) and 1(b)]. This can be described by the normal form of a tangent bifurcation, x˙ = x 2 − μ,
(13)
when choosing the parameter μ to be complex. The two complex fixed points can then be interchanged by encircling the exceptional point μ = 0 along the unit circle in the complex plane (μ = eiϕ , ϕ ∈ [0, 2π )). Similarly, the behavior of an EPn can be described by the normal form x˙ = x n − μ.
(14)
When encircling the exceptional point, the n fixed points permute and the initial situation is restored after n cycles. Depending on the path for the encircling, the cusp bifurcation can show the behavior of an EP2 or an EP3: By
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1
Im(xi)
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-1 0.0π 0.5π 1.0π 1.5π 2.0π ϕ
-1 0.0π 0.5π 1.0π 1.5π 2.0π ϕ (d)
6
E = -1.544 E = -1.000 E = 0.000 E = 0.356 E = 0.464
4
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-4 0.5π 1.0π 1.5π 2.0π ϕ
1
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4 Im(xi)
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PHYSICAL REVIEW E 91, 012915 (2015)
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III. RESULTS OF CLASSICAL CALCULATIONS AND DISCUSSION
Figure 2(a) shows the shape of the quasi-Penning resonances when following them up to higher energies at a fixed field strength. Since the name quasi-Penning was applied to almost elliptical orbits localized in the vicinity of the Stark saddle point and referred to the structural similarity between the equations of motion linearized around the saddle point and the stability conditions in a Penning trap, we will now refer to them more generally as B1 . The differences from the case of energies slightly above ESP are obvious: When increasing
0 1 x [scaled a.u.]
2
3
(b) B1, E = 1.0 B2, E = 1.0 D1, E = 1.3 K1, E = 1.0
4
FIG. 1. (Color online) By encircling an EP2, an interchange of two resonances can be observed. Parts (a) and (b) show the real and imaginary part of the two fixed points of Eq. (13), depending on the angle ϕ in μ = eiϕ . (c),(d) When encircling the cusp point along (a, b) = (−5 cos ϕ, −5 sin ϕ), ϕ ∈ [0, 2π ), one of the fixed points of Eq. (11) can be transformed into another one [marked by the red solid line starting in the upper left corner of (c)], without passing through a bifurcation. A second encircling passes both tangent bifurcations. Since the fixed points coincide in those bifurcations, it is not possible to determine the permutation behavior. An EP2 or an EP3 behavior can be observed accordingly. The angle ϕ and the position x are given in dimensionless units [see Eqs. (11) and (13)].
choosing b = 0 and a = eiϕ , one obtains an EP2 behavior. On the other hand, an EP3 behavior is obtained by setting a = 0 and b = eiϕ . This phenomenon was already observed in quantum-mechanical calculations for dipolar Bose-Einstein condensates [46,47]. The path of Eq. (12) with real parameters a and b gives rise to an ambiguity between both behaviors: If one traces two fixed points beyond a tangent bifurcation, two complex fixed points will appear. Since those two points coincide in the tangent bifurcation, it is not possible to relate one of the complex points to one of the real points, respectively. The permutation behavior cannot be determined, which is shown in Figs. 1(c) and 1(d). This ambiguity exists only for real parameter values. If a small complex part is added to a or b, an unambiguous determination of the permutation behavior is possible.
-1
y [scaled a.u.]
2
0
-2
-4 -4
-2
0
2
4
x [scaled a.u.]
FIG. 2. (Color online) (a) Shape of the orbit B1 , evolving from the elliptically shaped quasi-Penning resonances, at different energies and a fixed value of F = 0.6. The orbit is localized in the z = 0 plane. The position of the Stark saddle point on the left-hand side and the nucleus at the origin are marked by diamonds. At E ≈ 0.356 the right point of intersection with the x axis coincides with the nucleus. (b) Shape of the orbits B1 , B2 , D1 and K1 at F = 0.5. The positions x and y are given in scaled atomic units [see Eq. (2)].
the energy, the orbit becomes more and more heart-shaped. Figure 2(b) shows in advance the shortest orbits in the z = 0 plane, which are now investigated in detail. In the following figures, we indicate the stability of orbits by two upper indices added to the name of the orbit. The first one refers to the stability (s) or instability (u) perpendicular to the magnetic field. The second one indicates the stability parallel to the magnetic field. Regions of different stability are additionally dyed by colors, shadings, and grayscales. The stability behavior of the orbits B1 is shown in Fig. 3(a). After coming into existence, they are at first stable parallel to the magnetic field, since the Stark saddle point is a local potential minimum in this direction. Toward higher energies they become completely unstable and may vanish in bifurcations. We note that above a specific field strength of Fxy,BK = 0.481, the right point of intersection with the x axis
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FIG. 3. (Color online) Stability of B1 (a) and B2 (b) in dependence on the scaled energy and the scaled field strength. Along the lines marking the borders of different stability regions, bifurcations occur. The stability within those regions is displayed by two upper indices. The first one refers to the stability (s) or instability (u) of the orbit in the direction perpendicular to the magnetic field, the second one to the stability in the direction parallel to the magnetic field. According to the different types of stability, the regions are additionally dyed by colors, shadings, and grayscales. One can clearly observe the stability apex for B1 ending at F ≈ 0.5, E ≈ 0.5, i.e., a point in the E − F space at which two border lines of the stability regions B1us − B1uu meet, which was first uncovered by Fl¨othmann [38]. The dashed line on the left-hand side in (a) marks the energy of the saddle point (SP) as a function of the field strength. Specific points in parameter space are displayed by Si = (Fi , Ei ). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
of the orbits B1 coincides with the nucleus. This coincidence occurs along the dash-dotted black line in Fig. 3(a), which passes through the lower area of instability. Along this line, the orbits B1 are closed periodic orbits starting at and returning to the nucleus [71–73]. Afterward, a change of the shape takes place before the orbit vanishes in a pitchfork bifurcation with period-doubling along the right dashed line in Fig. 3(a). In crossed fields two orbits exist, which are localized in the z = 0 plane and which change over to the Kepler orbits [9,38] in the case of vanishing fields. We will refer to them as K1 and K2 . The second orbit, K2 , distinguishing itself from the first one by its sense of rotation around the nucleus, is not involved in any of the bifurcations considered and therefore not shown in Fig. 2(b). The lastly mentioned dashed red line marks also a stability change perpendicular to the magnetic field of K1 , which is related to a bifurcation between B1 and K1 . Beside the bifurcation with K1 , two other bifurcations can be observed for B1 : At the stability change in the z direction, pitchfork bifurcations with period-doubling occur in which three-dimensional orbits are created. Along the remaining solid blue line at high energies separating the white area from the area of instability in Fig. 3(a), a tangent bifurcation with B2 occurs. This orbit has a similar shape to that of B1 , but it exists only at high energies and field strengths [Fig. 3(b)]. If one takes a closer look at the stability areas according to the z direction, one can see that they end in an apex not only for B1 but also for B2 . Both apexes meet at the point Sz,B = (Fz,B , Ez,B ) = (0.510,0.548), which is located on the bifurcation line between B1 and B2 (Fig. 4). Therefore, we conclude that stability borders limiting the area of stability in the z direction for B1 continue as the according stability borders of B2 . Taking a closer look at this region of parameter space in Fig. 4(b), a cusp bifurcation between orbits B1 and B2 is observed. As described in Sec. II B, two tangent bifurcation lines coincide in the cusp point—here SCPB = (0.511,0.535)— without continuation. Along both lines, bifurcations between B1 and B2 occur. But while B2 exists only in the area between both lines, B1 exists in the complete external region and in the area between both lines twice. We will refer to those two different versions of B1 as B1a and B1b . The three orbits B1a , B1b , and B2 correspond to the three fixed points of the cusp bifurcation in Sec. II B. Starting from the lower continuous line on the right-hand side of SCPB , we follow the orbit B1a or B1 anticlockwise around the cusp point and again in the darker marked area until it vanishes as B1b in a tangent bifurcation along the dashed line between SCPB and Sxy,BK . Therefore, B1a and B1b can be converted into each other by encircling SCPB . This behavior is the same as for the fixed points of Eq. (11). By allowing the coordinates and the time to become complex, the analytically continued orbits can be followed beyond the tangent bifurcations. A plot of one of the intersection points of the orbits with the x axis versus the angle ϕ, which parametrizes the circle around SCPB , is shown in Fig. 5. The left tangent bifurcation line of B1 and B2 in Fig. 4(b) leading from SCPB to Sxy,BK coincides in Sxy,BK with the dashed pitchfork bifurcation line of B1 and K1 also ending in Sxy,BK . It continues itself toward lower field strengths as a pitchfork bifurcation line between B2 and K1 . This line marks the upper boundary of the region in which B2 exists. Examining the stability behavior of B2 , one can notice another
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Re(x0) [scaled a.u.]
(a)
0 -0.4
B1a B1
B1b B2
CX1 CX2
-0.8 -1.2 -1.6 -2 -2.4 0.0π
0.5π
1.0π ϕ
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CLASSICAL DYNAMICS AND LOCALIZATION OF . . .
0.8
B1a B1
B1b B2
CX1 CX2
0.4 0 -0.4 -0.8 0.0π
0.5π
1.0π
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2.0π
ϕ
FIG. 5. (Color online) Real (a) and imaginary part (b) of one of the intersection points of the orbits with the x axis when encircling the cusp point SCPB . The angle ϕ parametrizes the circle around SCPB . CX1 and CX2 are the two complex orbits appearing. Just as in Figs. 1(c) and 1(d), it is not possible to determine the permutation behavior due to the coincidence of two orbits in a tangent bifurcation, respectively. The angle ϕ and the values of x0 are given in dimensionless and scaled atomic units, respectively [see Eq. (2)].
describing a stability change of B2 perpendicular to the magnetic field, a pitchfork bifurcation with period-doubling occurs. The period-doubled orbit coming into being is named D1 (compare Fig. 2). For D1 both of the described phenomena can be found again. In Fig. 7(b), a very small region of complete stability localized at even higher energies than the stability island of B2 is displayed. The cusp bifurcation occurs around the cusp point SCPD , in which again two tangent bifurcation lines coincide without continuation. The darker marked areas denote the existence of two different versions of D1 : D1a and D1b . We conclude that the reappearance of those phenomena indicates the possibility to find them for other orbits of even more complicated shape on-and-off-again. IV. SEMICLASSICAL QUANTIZATION AND EXACT QUANTUM-MECHANICAL CALCULATIONS
According to periodic orbit theory, every classical periodic orbit causes a modulation in the density of states [48–57]. Oscillating modulations can also be found in photoabsorption spectra or action spectra, respectively, and according to Gutzwiller [49] and Miller [58] they are located at FIG. 4. (Color online) Stability of B1 in dependence on the scaled energy and the scaled field strength. Part (a) is an enlargement of the rectangular area marked by dashed lines in Fig. 3(a). Part (b) is an enlargement of the equivalently marked area in (a). The different resolutions allow for a comparison with Fig. 6 and provide better insight in the relevant region around F ≈ 0.5, E ≈ 0.6. In the dark red area, two different versions of B1 exist, which are indicated by B1a and B1b . The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
interesting phenomenon: The orbit becomes completely stable in a relatively large area of parameter space (green area marked B2ss in Fig. 6). This is used as an opportunity to carry out a semiclassical quantization of B2 and to search for signatures in exact quantum spectra in Sec. V. Finally, one last orbit localized in the z = 0 plane is worth mentioning. Along the dash-dotted blue line in Fig. 3(b)
2 1 λ ϕi = 2π n + , mi + S− 2 4 i=1
(15)
with the action S, the stability angles ϕi = arg(di ), determined by the eigenvalues di with |di | 1 of the monodromy matrix, indicating the stability parallel () and perpendicular (⊥) to the magnetic field, the Maslov index λ, and the quantum numbers n, m1 , m2 . Those quantum numbers count the number of quanta along (n) and perpendicular (mi ) to the periodic orbit. Stable periodic orbits cause modulations of the density of states, which can be described by δ-function peaks, while unstable orbits cause broadened peaks. Those broadened peaks have the shape of Lorentzians in action spectra, i.e., when they are plotted against the action S. In the following, we will use the term “semiclassical half-width” for the half-widths of those peaks according to energy. For each value of n and each constant value of the field strengths, the widths according to
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FIG. 6. (Color online) (a) Stability of B2 in dependence on the scaled energy and the scaled field strength. The observed stability island is triangular in shape, indicated by B2ss , and it ends in Sz,B . Part (b) is an enlargement of the equivalently marked area in (a). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
energy can be calculated as the difference between the two points, where 2 1 λ ϕi = 2π n + mi + ln|di | − S± 2 4 i=1 i=1 2
FIG. 7. (Color online) Stability of D1 in dependence on the scaled energy and the scaled field strength. A very thin region marked D1ss of complete stability can be observed. A cusp bifurcation occurs around the cusp point SCPD . Part (b) is an enlargement of the equivalently marked area in (a). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
The quantum-mechanical resonance spectra and wave functions are determined as described in [59]: The Schr¨odinger equation with the unscaled Hamiltonian (1) is rewritten in dilated semiparabolic coordinates [37,60]
(16)
is fulfilled [49]. Since we did not calculate the Maslov index of B2 , we assume in the following λ = 1, which will supply the expected agreement between semiclassical and quantummechanical results.
μ=
1√ y 1√ r + z, ν = r − z, and ϕ = arctan . b b x
(17)
Note that we cannot additionally use scaled variables in quantum-mechanical calculations since the scaling is restricted to the classical dynamics [19,38]. The parameter b = |b|eiϑ/2 introduces a complex delatation of the coordinates r, which
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is necessary to determine resonances using the complex coordinate rotation method [61–63]. The Schr¨odinger equation then reads ∂ μ + ν − (μ2 + ν 2 ) + 4b2 + b4 B(μ2 + ν 2 )i ∂ϕ 1 − b8 B 2 μ2 ν 2 (μ2 + ν 2 ) − 2b6 F μν(μ2 + ν 2 ) cos ϕ ψ 4 = [ (μ2 + ν 2 )]ψ,
(18)
with the Laplacians ρ =
1 ∂2 1 ∂ ∂ ρ + 2 2 , ρ ∈ {μ, ν}, ρ ∂ρ ∂ρ ρ ∂ϕ
(19)
and generalized complex eigenvalues = −(1 + 2b4 E), related to the complex energies E of resonances, the real parts of which represent their energies and the imaginary parts their widths = −2 Im(E). To calculate resonances, a matrix representation of the Schr¨odinger equation (18) is diagonalized. We use the adequate complete basis |nμ , nν , m = |nμ , m ⊗ |nν , m ,
2
/2 |m|
ρ
|m|
L(n−|m|)/2 (ρ 2 ),
(21)
(22)
nμ ,nν ,m
i∗ (μ,ν,ϕ) =
ci nμ , nν , m ψn∗μ nν m (μ,ν,ϕ),
(24)
nμ ,nν ,m
obtained as a result of the matrix diagonalization, are normalized according to d 3 r i∗ (μ,ν,ϕ)j (μ,ν,ϕ) =b
6
∞
dμ 0
∞
dν 0
0
2π
×
∗
∗
d 4|| ()G (, U, P) , (27) 2
2
with a Gaussian of minimum uncertainty (U P = 1/2) 1 2 1 e− 2σ 2 (−U) −i[ P+ A(U)] , (σ 2 π )1/2
(28)
the vector potential
and the associated Laguerre polynomials Lαk (x). In our numerical calculations, the maximum number of states used is limited by the condition nμ + nν 60 due to the required computer memory. Since μ and ν are complex coordinates, all nonintrinsically complex parts must remain unconjugated in the case of a complex conjugation [63,64]. This means that ψn∗μ nν m is equal to ψnμ nν m except for the term eimϕ , which is replaced with e−imϕ . The eigenstates ci nμ ,nν ,m ψnμ nν m (μ,ν,ϕ), (23) i (μ,ν,ϕ) =
G(, U, P) =
with fnm (ρ) = e−ρ
which is a bijection assuring α = 0 as long as U2 0 holds. The Husimi distribution [67–70] then reads PH (U, P) = d 2 4||2 ()G(, U, P)
(20)
with the eigenstates |nρ , m of the two-dimensional harmonic oscillator. The position space representation reads ψnμ nν m (μ,ν,ϕ) = μ,ν,ϕ|nμ ,nν ,m [(nμ − |m|)/2]![(nν − |m|)/2]! = [(nμ + |m|)/2]![(nν + |m|)/2]! 2 × fn m (μ)fnν m (ν)eimϕ , π μ
Due to the restriction of the complex conjugation to the intrinsically complex parts, the expression j∗ j is not a real quantity. The probability density is obtained as = |j∗ j | [65] instead. Finally, Husimi distributions can be calculated, allowing a comparison between the classical torus structure existing around the periodic orbits and quantum-Poincar´e sections [66]. To prevent a divergence of the momenta, we use regularized coordinates for the Husimi distributions, too. We will show that the probability density of the observed resonances is mainly limited to the z = 0 plane. For that reason, the calculations can be restricted to the x and y or U1 and U2 coordinates, respectively [cf. Eqs. (6a) and (10)]. By setting z = 0, one obtains 2 x U1 − U22 = , (26) y 2U1 U2
dϕ μν(μ2 + ν 2 )i∗ j = δij . (25)
−U2 1 , A = P + B U12 + U22 U1 2
(29)
and √ an appropriate adaptable squeezing parameter σ = U/P . To determine quantum-Poincar´e sections, U1 is set to the constant value U1 = 0. The conjugate momentum P1 is then calculated according to Eq. (7). V. RESULTS OF QUANTUM-MECHANICAL CALCULATIONS AND DISCUSSION
Since the classical calculations have been carried out for scaled energy values and scaled field strengths, one of the three parameters E, F , and B can now be chosen arbitrarily in quantum-mechanical calculations. In the following, we will set B to fixed values while the other parameters are calculated according to Eq. (2) and the location of the stability island. Figure 8 shows results at B = 1000 T. The color bar (grayscales) of this figure shows the quantum numbers n obtained by semiclassical quantization of B2 according to Eq. (15). The stability island is marked by dashed lines. Green (solid) lines within the stability island and blue (solid) lines outside of it display positions where integer values of n are obtained. White (solid) lines of parabolic shape outside the stability island display the semiclassical half-widths calculated by Eq. (16). For each constant value of F , the semiclassical half-width (according to energy) of a semiclassical state with quantum number n can be read out as the sum of the distances between the blue (solid) line for this value of n and the neighboring white (solid) lines. The real parts of the quantum-mechanical resonances (calculated only at discrete
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FIG. 8. (Color online) Comparison between quantumˆ T). mechanical and semiclassical results at B = 4.26 × 10−3 (=1000 For further information, see Sec. V. Best agreements are obtained for λ = 1 and m⊥ = 0, m = 0 in the case of the first resonance series (a) and m⊥ = 0, m = 2 in the case of the second resonance series (b). The parameter range shown is almost the same as in Fig. 6(a). All values are given in atomic units.
field strengths) are displayed by diamonds, the imaginary parts by error bars. Two series of resonances were found, which can be traced down to lower energies. It is supposed that this retracing will end in the resonances described in [59]. The contradictory fact that the resonances can also be found outside the region where the classical orbits B2 exist is an indication that a simple semiclassical quantization including solely B2 may be insufficient (cf. also Sec. VI). Semiclassical quantization normally includes all orbits appearing at a specific point in parameter space [48]. However, we assume that the resonances found within the area of B2 are closely related to those orbits. The energetic distance between two resonances according to the real part of the energy is almost the same within both series, while the widths differ strongly between the two series. Assuming that the resonances of smaller widths (first series) represent a quantum-mechanical ground state and the other
ones (second series) an excited state, semiclassical calculations are carried out to the quantum numbers m⊥ = 0,m = 0 and m⊥ = 0,m = 2, respectively. Since we consider only states of even parity in our quantum-mechanical calculations, the state with m⊥ = 0, m = 2 must be the first excited state appearing in the spectra. As concerns the positions of the resonances, a very good agreement between semiclassical and quantum-mechanical results is achieved outside the stability island. However, those quantum numbers lead to a switching of the semiclassical results from one resonance to another. Using different quantum numbers, an even worse agreement outside the stability island is obtained. We therefore conclude that m⊥ = 0 and m = 2 are correct and that the quantummechanical resonances do not show the switching behavior since they are not able to resolve the semiclassical structure in this region, which we will explain in the following. On the other hand, this switching can be seen as a further hint that a simple semiclassical quantization of B2 is insufficient and that all the other orbits have to be considered as well. As can be seen from Fig. 8, a good agreement according to the widths of the resonances is achieved only far away from the stability island. The expected decline of resonance widths within the stability island, proving the existence of quantum-mechanical bound states related to the classical stable orbits B2 , is not observed and will only show up at lower magnetic field strengths. When calculating Poincar´e sections of the classical torus structure existing around the periodic orbit, one uncovers that this structure is very small, too small to be resolved in quantum-mechanical calculations at B = 1000 T. Due to the expansion of this structure with a decreasing value of B (note that r = r˜ B −2/3 ), one expects a better resolution at lower field strengths. The increasing quantum number of n makes it necessary to increase the number of basis functions used in order to ensure a convergence of the quantum-mechanical calculations. Within the limitation of nμ + nν 60, it is possible to follow the resonances down to B = 40 T. Figure 9 shows that even at this field strength, the resonance widths do not vanish within the stability island. The probability densities (Figs. 10 and 11) exhibit instead a partially good agreement with the course of the classical orbit. Differences can be explained by the fact that the calculated wave function is only a snapshot while the resonance itself is a time evolving and finally decaying state. The higher quantum-mechanical probability density along the negative x axis possibly indicates the decay of the resonance since—from a classical point of view—all trajectories of the electron describing the ionization of the hydrogen atom have to pass the vicinity of the Stark saddle point [59]. This higher probability density then shows up again along the U2 axis in Fig. 12 [and it will finally be seen as a displacement between classical and quantum-mechanical structure in Fig. 13(a)]. It can be observed that the larger values of the probability density are taken on in the z = 0 plane when regarding the first resonance series. In the case of the second resonance series, the extension in the z direction is much larger. The calculated quantum-Poincar´e sections are compared with the classical ones in Fig. 13. As expected, the torus structure is, especially in the direction of the momentum P2 , much smaller than the quantum-mechanical structure.
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FIG. 9. (Color online) Comparison between semiclassical and quantum-mechanical results for the first resonance series at B = ˆ T). The resonance widths are evidently smaller but 1.70 × 10−4 (=40 still do not vanish within the stability island, which is marked by dashed lines. For further information, see Sec. V. All values are given in atomic units.
FIG. 11. (Color online) Probability density (r) of a resonance of the second series at E = (2.14 × 10−3 ) − (5.87 × 10−5 i), F = 4.73 × 10−6 , and B = 1.70 × 10−4 . The differences to the classical orbit (continuous line) are significantly larger. All values are given in atomic units.
FIG. 10. (Color online) Probability density (r) of a resonance of the first series at E = (1.95 × 10−3 ) − (4.14 × 10−5 i), F = 4.76 × 10−6 , and B = 1.70 × 10−4 . One can observe an agreement with the course of the classical orbit (continuous line) in (a) and the restriction to the z = 0 plane in (b). All values are given in atomic units.
FIG. 12. (Color online) Probability density |4U 2 ∗ (U)(U)| of the resonance of Fig. 10 in regularized coordinates. Without the limitation of U2 0, the probability density shows a second symmetry relative to the line U2 = 0. The lower part (U2 0) corresponds to α = 2π and is not included in the calculations of the quantum-Poincar´e sections. The classical orbit intersects the U1 = 0 line twice but with a different sign of the velocity V1 in the U1 direction. All values are given in atomic units.
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of 12 , we obtain a coincidence not until B ≈ 90 mT. This field strength is convenient for an experimental proof of the orbits B2 in photoabsorption spectra. As has been described, one of the intersection points of the classical orbits with the x axis is very close to the nucleus within the relevant region around F ≈ 0.5, E ≈ 0.6. An existing overlap of the corresponding resonances and the lowest states (1s or 2p) of the hydrogen atom allows, therefore, an experimental excitation of the hydrogen atom into those resonance states. According to closed orbit theory [71–73] and periodic orbit theory [49–57], the classical periodic orbits then become noticeable in photoabsorption spectra by oscillating terms ˜ −1/3 + ϕ), in which the amplitudes of the form A sin(2π SB depend on the stability of the orbits as well as the process of excitation. For the purpose of experimental proof, the data of a classical orbit located in the center of the stability island are given in scaled atomic units: F˜ = 0.4985, E˜ = 0.75, and S˜ = 14.7843. The conversion to unscaled units at the chosen magnetic field strength is obtained by Eq. (2) and S˜ = SB 1/3 . VI. SUMMARY AND CONCLUSION
FIG. 13. (Color online) Comparison between quantum-Poincar´e sections [shown by color bar (grayscales)] and classical torus structure. The dashed lines display the borders of the classically permitted region. Since the Hamiltonian is a quadratic function of the momentum, there are two possibilities to choose P1 when calculating the section. Both solutions differ from each other in the sign of the velocity V1 in the U1 direction. Therefore, agreements with only one part of the torus structure can be achieved at any time. The color bar is chosen equal to compare both results. We set σ to the constant value σ = 2. All values are given in atomic units.
Finally, we want to estimate at which field strengths a solution of the classical structure may be possible. Since the classical orbit is localized in the z = 0 plane, we regard the extension of the torus structure around the orbit in a z − pz -Poincar´e section as its classical uncertainty (note that we change back to nonregularized coordinates in this place). Assuming that the quantum mechanically determined uncertainty product zpz = 1.59 for one resonance of the first series does not change to a great extent with the value of B, since it is already sufficiently close to the critical value
Following the quasi-Penning resonances up to high energies and field strengths, we could resolve their complete bifurcation behavior. We found out that only K1 , B2 , and three-dimensional orbits are involved in the bifurcations of B1 . It was shown that several phenomena appear in the region of F ≈ 0.5, E ≈ 0.6 including a cusp bifurcation between B1 and B2 as well as the appearance of a stability island for B2 . The reappearance of those phenomena when analyzing the stability and bifurcation behavior of D1 could then be interpreted as a possibility to find them on-and-off-again for orbits of even more complicated shape. The puzzle of the z-stability region of B1 ending in an apex in parameter space could be resolved by a closer examination of the stability of B2 . The results show a coincidence of two stability apexes at Sz,B , which indicates the continuation of the stability borders limiting the areas of stability in the z direction from one of those orbits to the other one. Semiclassical quantizations of B2 showed agreements with resonances found in exact quantum spectra. Due to further agreements between quantum-mechanical probability densities and classical orbits as well as between quantum-Poincar´e sections and classical torus structure, we have to conclude that signatures of the stable orbit B2 in exact quantum spectra have been found. Due to limited computer memory and power, it is not yet possible to reach regions of several millitesla in order to find out whether the widths of the detected resonances will disappear within the stability island or not. Nevertheless, we think that an experimental proof of B2 in photoabsorption spectra at the estimated field strengths will eventually be possible. Finally, since the widths of the observed resonances do not vanish within the stability island of B2 , and since the stability island is very close to the bifurcation lines with B1a and B1b , we think that the orbits B2 cannot be treated as isolated ones in a semiclassical quantization. It would therefore be preferable to perform a uniform semiclassical quantization [74–76] of the observed cusp bifurcation.
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ACKNOWLEDGMENTS
We thank Eugen Fl¨othmann for stimulating discussions.
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Classical dynamics and localization of resonances in the high-energy region of the hydrogen atom in crossed fields Frank Schweiner, J¨org Main, Holger Cartarius, and G¨unter Wunner Institut f¨ur Theoretische Physik 1, Universit¨at Stuttgart, 70550 Stuttgart, Germany (Received 7 October 2014; published 20 January 2015) When superimposing the potentials of external fields on the Coulomb potential of the hydrogen atom, a saddle point (called the Stark saddle point) appears. For energies slightly above the saddle point energy, one can find classical orbits that are located in the vicinity of this point. We follow those so-called quasi-Penning orbits to high energies and field strengths, observing structural changes and uncovering their bifurcation behavior. By plotting the stability behavior of those orbits against energy and field strength, the appearance of a stability apex is reported. A cusp bifurcation, located in the vicinity of the apex, will be investigated in detail. In this cusp bifurcation, another orbit of similar shape is found. This orbit becomes completely stable in the observed region of positive energy, i.e., in a region of parameter space, where the Kepler-like orbits located around the nucleus are already unstable. By quantum mechanically exact calculations, we prove the existence of signatures in quantum spectra belonging to those orbits. Husimi distributions are used to compare quantum-Poincar´e sections with the extension of the classical torus structure around the orbits. Since periodic orbit theory predicts that each classical periodic orbit contributes an oscillating term to photoabsorption spectra, we finally give an estimation for future experiments, which could verify the existence of the stable orbits. DOI: 10.1103/PhysRevE.91.012915
PACS number(s): 32.60.+i, 32.80.−t, 05.45.−a
I. INTRODUCTION
The hydrogen atom in crossed electric and magnetic fields is a simple example of a nonintegrable physical system, which has been investigated for almost 100 years, theoretically [1–5] as well as experimentally [3–14]. Furthermore, it has been used for studies on phenomena such as Ericson fluctuations [8,15] or Arnol’d diffusion [6]. Even quantum dots [16] and excitons [17] in condensed-matter physics can be explored using the findings attained from the hydrogen atom in crossed fields. One of the major purposes in recent decades was to uncover how chaos, which can be observed in a classical treatment [18,19], shows itself in quantum spectra, since the Schr¨odinger equation is, due to its linearity, not capable of producing chaotic behavior [5,20–26]. Nevertheless, new phenomena occur in crossed fields: the so called quasiPenning resonances [27]. From a classical point of view, those resonances describe a movement of the electron that is localized around a saddle point in the potential, namely the Stark saddle point. The possible appearance of wave functions localized far away from the nucleus led to a series of further investigations [27–32], since they can play an important role in the ionization process. Transition state theory predicts classical orbits localized around the Stark saddle point and states that all ionizing orbits have to pass in the vicinity of this point [33–36]. Even though Clark et al. [27] already proved their existence in 1985, it was not until 2009 that the first signatures of those orbits were found in calculated quantum spectra [37]. The classical stability behavior of the quasi-Penning orbits at high energies and field strengths was first investigated by Fl¨othmann in 1994 [38]. He uncovered a stability apex in parameter space if the stability change of the orbits is illustrated as a function of both parameters. It is the purpose of this paper to perform more precise calculations on the stability of those orbits and to uncover the bifurcation behavior as well as the processes taking place around the stability apex. We report the existence of a cusp 1539-3755/2015/91(1)/012915(11)
bifurcation, which appears in the vicinity of the stability apex and involves another orbit of similar shape. This orbit, becoming completely stable in a specific area of parameter space, is the starting point for semiclassical and quantummechanical calculations. We will demonstrate that signatures of those orbits at high energies and field strengths can be found in accurately calculated quantum spectra. The paper is organized as follows. In Sec. II, the system is introduced, and a scaling of parameters as well as a regularization of coordinates are carried out. A comparison between exceptional points and a classical cusp bifurcation is drawn. The classical stability of the quasi-Penning resonances and their bifurcation behavior at high energies and field strengths are presented in Sec. III. An introduction into the semiclassical and exact quantum-mechanical calculations performed is given in Sec. IV, before pertinent results are discussed in Sec. V. In Sec. VI, a short summary is given and conclusions are drawn. II. CLASSICAL CALCULATIONS A. Hamiltonian, monodromy matrix, and regularized coordinates
The Hamiltonian of a hydrogen atom in a constant electric field F = F eˆx and a constant magnetic field B = B eˆz reads in atomic units (Hartree units with lengths, energies, and electric and magnetic field strengths given in units of 5.29 × 10−11 m, 4.36 × 10−18 J, 5.14 × 1011 V/m, and 2.35 × 105 T, respectively) H =
1 2 1 1 1 p − + BLz + B 2 (x 2 + y 2 ) + F x, 2 r 2 8
(1)
with Lz = xpy − ypx . In the following, we shall take advantage of a scaling property of the classical Hamiltonian enabling us to deal with only two independent variables, i.e., the scaled
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energy and the scaled field strength. The other variables have to be scaled as well, so that the scaling transformations read E˜ = EB −2/3 , F˜ = F B −4/3 , r˜ = rB 2/3 , p˜ = pB −1/3 , t˜ = tB.
(4)
1 L(U) P, r together with a transformation of time, dt = 2r dτ.
(6b)
(6c)
The equations of motion (3) and (5) remain the same except for the replacements γ → = (U, P)T and t → τ . The Hamiltonian then reads H = 12 P 2 − [E − F (U1 U3 − U2 U4 )]U 2
+ 12 (U1 P2 − U2 P1 ) U32 + U42 + (U3 P4 − U4 P3 ) U12 + U22 (7) + 18 U 2 U12 + U22 U32 + U42 = 2, with U 2 = 4i=1 Ui2 . By integrating two further equations along with the Hamiltonian equations of motion, one obtains
(8)
(9)
The transformation (6) is not bijective. For this reason, the inversion, ⎞ ⎛√ r + z cos ϕ+α 2 ⎜ √r + z sin ϕ+α ⎟ ⎜ 2 ⎟ (10) U = ⎜√ ⎟, ⎝ r − z cos ϕ−α ⎠ 2 √ r − z sin ϕ−α 2 contains an additional parameter α, which, without loss of generality, we will set to the constant value α = 0. B. Cusp bifurcation and exceptional points
A cusp bifurcation, appearing in systems with at least two parameters a and b, is described by the normal form [42] x˙ =
(5)
¯ The eigenvalues of M(t) therefore indicate whether a variation of the initial conditions of a periodic orbit leads to exponential divergence or a trajectory remains in the vicinity of the periodic orbit for all times. In the case of a Hamiltonian system, the energy conservation leads to two variational directions not affecting the system’s behavior. The corresponding eigenvalues hence take on the value of 1. Omitting those directions, one obtains the so-called monodromy matrix M. To prevent a divergence of the momentum p near the nucleus, the Kustaanheimo-Stiefel regularization of coordinates [40,41] is used, ⎛ ⎞ U3 −U4 U1 −U2 1 ⎜U U3 U2 U1 ⎟ r = L(U)U = ⎝ 4 U, (6a) U2 −U3 −U4 ⎠ 2 U1 U2 −U1 −U4 U3 p=
d 1
S = P 2 + (U1 P2 − U2 P1 ) U32 + U42 dτ 2 + (U3 P4 − U4 P3 ) U12 + U22 .
(2b)
The Stark saddle point is characterized by the √ fixed-point condition γ˙ = 0, yielding its position r = (−1/ F , 0, 0)T SP √ and energy ESP = −2 F . The stability of orbits is investi¯ M ¯ describes in a linear gated using the stability matrix M. approximation the relative behavior of two trajectories γ (1) and γ (2) in time [39],
and it can be determined by its equation of motion ∂ 2H ¯ d ¯ 0 1 ¯ M= J M with J = , M(0) = 1. −1 0 dt ∂γ 2
d t = U 2, dτ
(2a)
Without further usage of the tilde sign, the Hamiltonian has the same structure as in Eq. (1) when setting B = 1. Defining γ = (r, p)T , we want to find periodic solutions of the Hamiltonian equations of motion, ∂H d 0 1 γ = J with J = . (3) −1 0 dt ∂γ
(1) ¯ γ (1) (t) − γ (2) (t) = M(t)[γ (0) − γ (2) (0)],
the periods and actions of the orbits:
4 3 x 27
+ ax + b.
(11)
4 We choose the factor in front of x 3 as 27 instead of 1 to simplify the results. When calculating the fixed points of Eq. (11), one finds three real solutions for a < 0 and |b| < (−a)3/2 . The two boundary lines of this area are tangent bifurcation lines, along which two fixed points coincide. In the remaining parameter space, only one real solution can be found. A specific attribute of the cusp bifurcation shows a similarity to exceptional points: Following one fixed point around the cusp point, which is located at (a, b) = (0, 0), along a circle,
(a, b) = (−r cos ϕ, − r sin ϕ) with ϕ ∈ [0, 2π ),
(12)
it can be transformed into another one. This phenomenon appears in the case of exceptional points, too. An exceptional point, first described by Kato [43], marks the coalescence of at least two resonances (or more precisely, eigenvalues and corresponding eigenvectors) of a complex Hamiltonian in an at least two-dimensional parameter space [44–47]. Encircling the exceptional point in parameter space, the resonances permute. In the case of an EP2, two resonances interchange and the initial situation can be restored after two cycles [Figs. 1(a) and 1(b)]. This can be described by the normal form of a tangent bifurcation, x˙ = x 2 − μ,
(13)
when choosing the parameter μ to be complex. The two complex fixed points can then be interchanged by encircling the exceptional point μ = 0 along the unit circle in the complex plane (μ = eiϕ , ϕ ∈ [0, 2π )). Similarly, the behavior of an EPn can be described by the normal form x˙ = x n − μ.
(14)
When encircling the exceptional point, the n fixed points permute and the initial situation is restored after n cycles. Depending on the path for the encircling, the cusp bifurcation can show the behavior of an EP2 or an EP3: By
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1
Im(xi)
Re(xi)
0
-1 0.0π 0.5π 1.0π 1.5π 2.0π ϕ
-1 0.0π 0.5π 1.0π 1.5π 2.0π ϕ (d)
6
E = -1.544 E = -1.000 E = 0.000 E = 0.356 E = 0.464
4
2
2
0
0
-1
0
-2
-2
-4
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4 Im(xi)
Re(xi)
3
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III. RESULTS OF CLASSICAL CALCULATIONS AND DISCUSSION
Figure 2(a) shows the shape of the quasi-Penning resonances when following them up to higher energies at a fixed field strength. Since the name quasi-Penning was applied to almost elliptical orbits localized in the vicinity of the Stark saddle point and referred to the structural similarity between the equations of motion linearized around the saddle point and the stability conditions in a Penning trap, we will now refer to them more generally as B1 . The differences from the case of energies slightly above ESP are obvious: When increasing
0 1 x [scaled a.u.]
2
3
(b) B1, E = 1.0 B2, E = 1.0 D1, E = 1.3 K1, E = 1.0
4
FIG. 1. (Color online) By encircling an EP2, an interchange of two resonances can be observed. Parts (a) and (b) show the real and imaginary part of the two fixed points of Eq. (13), depending on the angle ϕ in μ = eiϕ . (c),(d) When encircling the cusp point along (a, b) = (−5 cos ϕ, −5 sin ϕ), ϕ ∈ [0, 2π ), one of the fixed points of Eq. (11) can be transformed into another one [marked by the red solid line starting in the upper left corner of (c)], without passing through a bifurcation. A second encircling passes both tangent bifurcations. Since the fixed points coincide in those bifurcations, it is not possible to determine the permutation behavior. An EP2 or an EP3 behavior can be observed accordingly. The angle ϕ and the position x are given in dimensionless units [see Eqs. (11) and (13)].
choosing b = 0 and a = eiϕ , one obtains an EP2 behavior. On the other hand, an EP3 behavior is obtained by setting a = 0 and b = eiϕ . This phenomenon was already observed in quantum-mechanical calculations for dipolar Bose-Einstein condensates [46,47]. The path of Eq. (12) with real parameters a and b gives rise to an ambiguity between both behaviors: If one traces two fixed points beyond a tangent bifurcation, two complex fixed points will appear. Since those two points coincide in the tangent bifurcation, it is not possible to relate one of the complex points to one of the real points, respectively. The permutation behavior cannot be determined, which is shown in Figs. 1(c) and 1(d). This ambiguity exists only for real parameter values. If a small complex part is added to a or b, an unambiguous determination of the permutation behavior is possible.
-1
y [scaled a.u.]
2
0
-2
-4 -4
-2
0
2
4
x [scaled a.u.]
FIG. 2. (Color online) (a) Shape of the orbit B1 , evolving from the elliptically shaped quasi-Penning resonances, at different energies and a fixed value of F = 0.6. The orbit is localized in the z = 0 plane. The position of the Stark saddle point on the left-hand side and the nucleus at the origin are marked by diamonds. At E ≈ 0.356 the right point of intersection with the x axis coincides with the nucleus. (b) Shape of the orbits B1 , B2 , D1 and K1 at F = 0.5. The positions x and y are given in scaled atomic units [see Eq. (2)].
the energy, the orbit becomes more and more heart-shaped. Figure 2(b) shows in advance the shortest orbits in the z = 0 plane, which are now investigated in detail. In the following figures, we indicate the stability of orbits by two upper indices added to the name of the orbit. The first one refers to the stability (s) or instability (u) perpendicular to the magnetic field. The second one indicates the stability parallel to the magnetic field. Regions of different stability are additionally dyed by colors, shadings, and grayscales. The stability behavior of the orbits B1 is shown in Fig. 3(a). After coming into existence, they are at first stable parallel to the magnetic field, since the Stark saddle point is a local potential minimum in this direction. Toward higher energies they become completely unstable and may vanish in bifurcations. We note that above a specific field strength of Fxy,BK = 0.481, the right point of intersection with the x axis
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FIG. 3. (Color online) Stability of B1 (a) and B2 (b) in dependence on the scaled energy and the scaled field strength. Along the lines marking the borders of different stability regions, bifurcations occur. The stability within those regions is displayed by two upper indices. The first one refers to the stability (s) or instability (u) of the orbit in the direction perpendicular to the magnetic field, the second one to the stability in the direction parallel to the magnetic field. According to the different types of stability, the regions are additionally dyed by colors, shadings, and grayscales. One can clearly observe the stability apex for B1 ending at F ≈ 0.5, E ≈ 0.5, i.e., a point in the E − F space at which two border lines of the stability regions B1us − B1uu meet, which was first uncovered by Fl¨othmann [38]. The dashed line on the left-hand side in (a) marks the energy of the saddle point (SP) as a function of the field strength. Specific points in parameter space are displayed by Si = (Fi , Ei ). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
of the orbits B1 coincides with the nucleus. This coincidence occurs along the dash-dotted black line in Fig. 3(a), which passes through the lower area of instability. Along this line, the orbits B1 are closed periodic orbits starting at and returning to the nucleus [71–73]. Afterward, a change of the shape takes place before the orbit vanishes in a pitchfork bifurcation with period-doubling along the right dashed line in Fig. 3(a). In crossed fields two orbits exist, which are localized in the z = 0 plane and which change over to the Kepler orbits [9,38] in the case of vanishing fields. We will refer to them as K1 and K2 . The second orbit, K2 , distinguishing itself from the first one by its sense of rotation around the nucleus, is not involved in any of the bifurcations considered and therefore not shown in Fig. 2(b). The lastly mentioned dashed red line marks also a stability change perpendicular to the magnetic field of K1 , which is related to a bifurcation between B1 and K1 . Beside the bifurcation with K1 , two other bifurcations can be observed for B1 : At the stability change in the z direction, pitchfork bifurcations with period-doubling occur in which three-dimensional orbits are created. Along the remaining solid blue line at high energies separating the white area from the area of instability in Fig. 3(a), a tangent bifurcation with B2 occurs. This orbit has a similar shape to that of B1 , but it exists only at high energies and field strengths [Fig. 3(b)]. If one takes a closer look at the stability areas according to the z direction, one can see that they end in an apex not only for B1 but also for B2 . Both apexes meet at the point Sz,B = (Fz,B , Ez,B ) = (0.510,0.548), which is located on the bifurcation line between B1 and B2 (Fig. 4). Therefore, we conclude that stability borders limiting the area of stability in the z direction for B1 continue as the according stability borders of B2 . Taking a closer look at this region of parameter space in Fig. 4(b), a cusp bifurcation between orbits B1 and B2 is observed. As described in Sec. II B, two tangent bifurcation lines coincide in the cusp point—here SCPB = (0.511,0.535)— without continuation. Along both lines, bifurcations between B1 and B2 occur. But while B2 exists only in the area between both lines, B1 exists in the complete external region and in the area between both lines twice. We will refer to those two different versions of B1 as B1a and B1b . The three orbits B1a , B1b , and B2 correspond to the three fixed points of the cusp bifurcation in Sec. II B. Starting from the lower continuous line on the right-hand side of SCPB , we follow the orbit B1a or B1 anticlockwise around the cusp point and again in the darker marked area until it vanishes as B1b in a tangent bifurcation along the dashed line between SCPB and Sxy,BK . Therefore, B1a and B1b can be converted into each other by encircling SCPB . This behavior is the same as for the fixed points of Eq. (11). By allowing the coordinates and the time to become complex, the analytically continued orbits can be followed beyond the tangent bifurcations. A plot of one of the intersection points of the orbits with the x axis versus the angle ϕ, which parametrizes the circle around SCPB , is shown in Fig. 5. The left tangent bifurcation line of B1 and B2 in Fig. 4(b) leading from SCPB to Sxy,BK coincides in Sxy,BK with the dashed pitchfork bifurcation line of B1 and K1 also ending in Sxy,BK . It continues itself toward lower field strengths as a pitchfork bifurcation line between B2 and K1 . This line marks the upper boundary of the region in which B2 exists. Examining the stability behavior of B2 , one can notice another
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CX1 CX2
0.4 0 -0.4 -0.8 0.0π
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FIG. 5. (Color online) Real (a) and imaginary part (b) of one of the intersection points of the orbits with the x axis when encircling the cusp point SCPB . The angle ϕ parametrizes the circle around SCPB . CX1 and CX2 are the two complex orbits appearing. Just as in Figs. 1(c) and 1(d), it is not possible to determine the permutation behavior due to the coincidence of two orbits in a tangent bifurcation, respectively. The angle ϕ and the values of x0 are given in dimensionless and scaled atomic units, respectively [see Eq. (2)].
describing a stability change of B2 perpendicular to the magnetic field, a pitchfork bifurcation with period-doubling occurs. The period-doubled orbit coming into being is named D1 (compare Fig. 2). For D1 both of the described phenomena can be found again. In Fig. 7(b), a very small region of complete stability localized at even higher energies than the stability island of B2 is displayed. The cusp bifurcation occurs around the cusp point SCPD , in which again two tangent bifurcation lines coincide without continuation. The darker marked areas denote the existence of two different versions of D1 : D1a and D1b . We conclude that the reappearance of those phenomena indicates the possibility to find them for other orbits of even more complicated shape on-and-off-again. IV. SEMICLASSICAL QUANTIZATION AND EXACT QUANTUM-MECHANICAL CALCULATIONS
According to periodic orbit theory, every classical periodic orbit causes a modulation in the density of states [48–57]. Oscillating modulations can also be found in photoabsorption spectra or action spectra, respectively, and according to Gutzwiller [49] and Miller [58] they are located at FIG. 4. (Color online) Stability of B1 in dependence on the scaled energy and the scaled field strength. Part (a) is an enlargement of the rectangular area marked by dashed lines in Fig. 3(a). Part (b) is an enlargement of the equivalently marked area in (a). The different resolutions allow for a comparison with Fig. 6 and provide better insight in the relevant region around F ≈ 0.5, E ≈ 0.6. In the dark red area, two different versions of B1 exist, which are indicated by B1a and B1b . The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
interesting phenomenon: The orbit becomes completely stable in a relatively large area of parameter space (green area marked B2ss in Fig. 6). This is used as an opportunity to carry out a semiclassical quantization of B2 and to search for signatures in exact quantum spectra in Sec. V. Finally, one last orbit localized in the z = 0 plane is worth mentioning. Along the dash-dotted blue line in Fig. 3(b)
2 1 λ ϕi = 2π n + , mi + S− 2 4 i=1
(15)
with the action S, the stability angles ϕi = arg(di ), determined by the eigenvalues di with |di | 1 of the monodromy matrix, indicating the stability parallel () and perpendicular (⊥) to the magnetic field, the Maslov index λ, and the quantum numbers n, m1 , m2 . Those quantum numbers count the number of quanta along (n) and perpendicular (mi ) to the periodic orbit. Stable periodic orbits cause modulations of the density of states, which can be described by δ-function peaks, while unstable orbits cause broadened peaks. Those broadened peaks have the shape of Lorentzians in action spectra, i.e., when they are plotted against the action S. In the following, we will use the term “semiclassical half-width” for the half-widths of those peaks according to energy. For each value of n and each constant value of the field strengths, the widths according to
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FIG. 6. (Color online) (a) Stability of B2 in dependence on the scaled energy and the scaled field strength. The observed stability island is triangular in shape, indicated by B2ss , and it ends in Sz,B . Part (b) is an enlargement of the equivalently marked area in (a). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
energy can be calculated as the difference between the two points, where 2 1 λ ϕi = 2π n + mi + ln|di | − S± 2 4 i=1 i=1 2
FIG. 7. (Color online) Stability of D1 in dependence on the scaled energy and the scaled field strength. A very thin region marked D1ss of complete stability can be observed. A cusp bifurcation occurs around the cusp point SCPD . Part (b) is an enlargement of the equivalently marked area in (a). The field strength F and the energy E are given in scaled atomic units [see Eq. (2)].
The quantum-mechanical resonance spectra and wave functions are determined as described in [59]: The Schr¨odinger equation with the unscaled Hamiltonian (1) is rewritten in dilated semiparabolic coordinates [37,60]
(16)
is fulfilled [49]. Since we did not calculate the Maslov index of B2 , we assume in the following λ = 1, which will supply the expected agreement between semiclassical and quantummechanical results.
μ=
1√ y 1√ r + z, ν = r − z, and ϕ = arctan . b b x
(17)
Note that we cannot additionally use scaled variables in quantum-mechanical calculations since the scaling is restricted to the classical dynamics [19,38]. The parameter b = |b|eiϑ/2 introduces a complex delatation of the coordinates r, which
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is necessary to determine resonances using the complex coordinate rotation method [61–63]. The Schr¨odinger equation then reads ∂ μ + ν − (μ2 + ν 2 ) + 4b2 + b4 B(μ2 + ν 2 )i ∂ϕ 1 − b8 B 2 μ2 ν 2 (μ2 + ν 2 ) − 2b6 F μν(μ2 + ν 2 ) cos ϕ ψ 4 = [ (μ2 + ν 2 )]ψ,
(18)
with the Laplacians ρ =
1 ∂2 1 ∂ ∂ ρ + 2 2 , ρ ∈ {μ, ν}, ρ ∂ρ ∂ρ ρ ∂ϕ
(19)
and generalized complex eigenvalues = −(1 + 2b4 E), related to the complex energies E of resonances, the real parts of which represent their energies and the imaginary parts their widths = −2 Im(E). To calculate resonances, a matrix representation of the Schr¨odinger equation (18) is diagonalized. We use the adequate complete basis |nμ , nν , m = |nμ , m ⊗ |nν , m ,
2
/2 |m|
ρ
|m|
L(n−|m|)/2 (ρ 2 ),
(21)
(22)
nμ ,nν ,m
i∗ (μ,ν,ϕ) =
ci nμ , nν , m ψn∗μ nν m (μ,ν,ϕ),
(24)
nμ ,nν ,m
obtained as a result of the matrix diagonalization, are normalized according to d 3 r i∗ (μ,ν,ϕ)j (μ,ν,ϕ) =b
6
∞
dμ 0
∞
dν 0
0
2π
×
∗
∗
d 4|| ()G (, U, P) , (27) 2
2
with a Gaussian of minimum uncertainty (U P = 1/2) 1 2 1 e− 2σ 2 (−U) −i[ P+ A(U)] , (σ 2 π )1/2
(28)
the vector potential
and the associated Laguerre polynomials Lαk (x). In our numerical calculations, the maximum number of states used is limited by the condition nμ + nν 60 due to the required computer memory. Since μ and ν are complex coordinates, all nonintrinsically complex parts must remain unconjugated in the case of a complex conjugation [63,64]. This means that ψn∗μ nν m is equal to ψnμ nν m except for the term eimϕ , which is replaced with e−imϕ . The eigenstates ci nμ ,nν ,m ψnμ nν m (μ,ν,ϕ), (23) i (μ,ν,ϕ) =
G(, U, P) =
with fnm (ρ) = e−ρ
which is a bijection assuring α = 0 as long as U2 0 holds. The Husimi distribution [67–70] then reads PH (U, P) = d 2 4||2 ()G(, U, P)
(20)
with the eigenstates |nρ , m of the two-dimensional harmonic oscillator. The position space representation reads ψnμ nν m (μ,ν,ϕ) = μ,ν,ϕ|nμ ,nν ,m [(nμ − |m|)/2]![(nν − |m|)/2]! = [(nμ + |m|)/2]![(nν + |m|)/2]! 2 × fn m (μ)fnν m (ν)eimϕ , π μ
Due to the restriction of the complex conjugation to the intrinsically complex parts, the expression j∗ j is not a real quantity. The probability density is obtained as = |j∗ j | [65] instead. Finally, Husimi distributions can be calculated, allowing a comparison between the classical torus structure existing around the periodic orbits and quantum-Poincar´e sections [66]. To prevent a divergence of the momenta, we use regularized coordinates for the Husimi distributions, too. We will show that the probability density of the observed resonances is mainly limited to the z = 0 plane. For that reason, the calculations can be restricted to the x and y or U1 and U2 coordinates, respectively [cf. Eqs. (6a) and (10)]. By setting z = 0, one obtains 2 x U1 − U22 = , (26) y 2U1 U2
dϕ μν(μ2 + ν 2 )i∗ j = δij . (25)
−U2 1 , A = P + B U12 + U22 U1 2
(29)
and √ an appropriate adaptable squeezing parameter σ = U/P . To determine quantum-Poincar´e sections, U1 is set to the constant value U1 = 0. The conjugate momentum P1 is then calculated according to Eq. (7). V. RESULTS OF QUANTUM-MECHANICAL CALCULATIONS AND DISCUSSION
Since the classical calculations have been carried out for scaled energy values and scaled field strengths, one of the three parameters E, F , and B can now be chosen arbitrarily in quantum-mechanical calculations. In the following, we will set B to fixed values while the other parameters are calculated according to Eq. (2) and the location of the stability island. Figure 8 shows results at B = 1000 T. The color bar (grayscales) of this figure shows the quantum numbers n obtained by semiclassical quantization of B2 according to Eq. (15). The stability island is marked by dashed lines. Green (solid) lines within the stability island and blue (solid) lines outside of it display positions where integer values of n are obtained. White (solid) lines of parabolic shape outside the stability island display the semiclassical half-widths calculated by Eq. (16). For each constant value of F , the semiclassical half-width (according to energy) of a semiclassical state with quantum number n can be read out as the sum of the distances between the blue (solid) line for this value of n and the neighboring white (solid) lines. The real parts of the quantum-mechanical resonances (calculated only at discrete
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FIG. 8. (Color online) Comparison between quantumˆ T). mechanical and semiclassical results at B = 4.26 × 10−3 (=1000 For further information, see Sec. V. Best agreements are obtained for λ = 1 and m⊥ = 0, m = 0 in the case of the first resonance series (a) and m⊥ = 0, m = 2 in the case of the second resonance series (b). The parameter range shown is almost the same as in Fig. 6(a). All values are given in atomic units.
field strengths) are displayed by diamonds, the imaginary parts by error bars. Two series of resonances were found, which can be traced down to lower energies. It is supposed that this retracing will end in the resonances described in [59]. The contradictory fact that the resonances can also be found outside the region where the classical orbits B2 exist is an indication that a simple semiclassical quantization including solely B2 may be insufficient (cf. also Sec. VI). Semiclassical quantization normally includes all orbits appearing at a specific point in parameter space [48]. However, we assume that the resonances found within the area of B2 are closely related to those orbits. The energetic distance between two resonances according to the real part of the energy is almost the same within both series, while the widths differ strongly between the two series. Assuming that the resonances of smaller widths (first series) represent a quantum-mechanical ground state and the other
ones (second series) an excited state, semiclassical calculations are carried out to the quantum numbers m⊥ = 0,m = 0 and m⊥ = 0,m = 2, respectively. Since we consider only states of even parity in our quantum-mechanical calculations, the state with m⊥ = 0, m = 2 must be the first excited state appearing in the spectra. As concerns the positions of the resonances, a very good agreement between semiclassical and quantum-mechanical results is achieved outside the stability island. However, those quantum numbers lead to a switching of the semiclassical results from one resonance to another. Using different quantum numbers, an even worse agreement outside the stability island is obtained. We therefore conclude that m⊥ = 0 and m = 2 are correct and that the quantummechanical resonances do not show the switching behavior since they are not able to resolve the semiclassical structure in this region, which we will explain in the following. On the other hand, this switching can be seen as a further hint that a simple semiclassical quantization of B2 is insufficient and that all the other orbits have to be considered as well. As can be seen from Fig. 8, a good agreement according to the widths of the resonances is achieved only far away from the stability island. The expected decline of resonance widths within the stability island, proving the existence of quantum-mechanical bound states related to the classical stable orbits B2 , is not observed and will only show up at lower magnetic field strengths. When calculating Poincar´e sections of the classical torus structure existing around the periodic orbit, one uncovers that this structure is very small, too small to be resolved in quantum-mechanical calculations at B = 1000 T. Due to the expansion of this structure with a decreasing value of B (note that r = r˜ B −2/3 ), one expects a better resolution at lower field strengths. The increasing quantum number of n makes it necessary to increase the number of basis functions used in order to ensure a convergence of the quantum-mechanical calculations. Within the limitation of nμ + nν 60, it is possible to follow the resonances down to B = 40 T. Figure 9 shows that even at this field strength, the resonance widths do not vanish within the stability island. The probability densities (Figs. 10 and 11) exhibit instead a partially good agreement with the course of the classical orbit. Differences can be explained by the fact that the calculated wave function is only a snapshot while the resonance itself is a time evolving and finally decaying state. The higher quantum-mechanical probability density along the negative x axis possibly indicates the decay of the resonance since—from a classical point of view—all trajectories of the electron describing the ionization of the hydrogen atom have to pass the vicinity of the Stark saddle point [59]. This higher probability density then shows up again along the U2 axis in Fig. 12 [and it will finally be seen as a displacement between classical and quantum-mechanical structure in Fig. 13(a)]. It can be observed that the larger values of the probability density are taken on in the z = 0 plane when regarding the first resonance series. In the case of the second resonance series, the extension in the z direction is much larger. The calculated quantum-Poincar´e sections are compared with the classical ones in Fig. 13. As expected, the torus structure is, especially in the direction of the momentum P2 , much smaller than the quantum-mechanical structure.
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FIG. 9. (Color online) Comparison between semiclassical and quantum-mechanical results for the first resonance series at B = ˆ T). The resonance widths are evidently smaller but 1.70 × 10−4 (=40 still do not vanish within the stability island, which is marked by dashed lines. For further information, see Sec. V. All values are given in atomic units.
FIG. 11. (Color online) Probability density (r) of a resonance of the second series at E = (2.14 × 10−3 ) − (5.87 × 10−5 i), F = 4.73 × 10−6 , and B = 1.70 × 10−4 . The differences to the classical orbit (continuous line) are significantly larger. All values are given in atomic units.
FIG. 10. (Color online) Probability density (r) of a resonance of the first series at E = (1.95 × 10−3 ) − (4.14 × 10−5 i), F = 4.76 × 10−6 , and B = 1.70 × 10−4 . One can observe an agreement with the course of the classical orbit (continuous line) in (a) and the restriction to the z = 0 plane in (b). All values are given in atomic units.
FIG. 12. (Color online) Probability density |4U 2 ∗ (U)(U)| of the resonance of Fig. 10 in regularized coordinates. Without the limitation of U2 0, the probability density shows a second symmetry relative to the line U2 = 0. The lower part (U2 0) corresponds to α = 2π and is not included in the calculations of the quantum-Poincar´e sections. The classical orbit intersects the U1 = 0 line twice but with a different sign of the velocity V1 in the U1 direction. All values are given in atomic units.
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of 12 , we obtain a coincidence not until B ≈ 90 mT. This field strength is convenient for an experimental proof of the orbits B2 in photoabsorption spectra. As has been described, one of the intersection points of the classical orbits with the x axis is very close to the nucleus within the relevant region around F ≈ 0.5, E ≈ 0.6. An existing overlap of the corresponding resonances and the lowest states (1s or 2p) of the hydrogen atom allows, therefore, an experimental excitation of the hydrogen atom into those resonance states. According to closed orbit theory [71–73] and periodic orbit theory [49–57], the classical periodic orbits then become noticeable in photoabsorption spectra by oscillating terms ˜ −1/3 + ϕ), in which the amplitudes of the form A sin(2π SB depend on the stability of the orbits as well as the process of excitation. For the purpose of experimental proof, the data of a classical orbit located in the center of the stability island are given in scaled atomic units: F˜ = 0.4985, E˜ = 0.75, and S˜ = 14.7843. The conversion to unscaled units at the chosen magnetic field strength is obtained by Eq. (2) and S˜ = SB 1/3 . VI. SUMMARY AND CONCLUSION
FIG. 13. (Color online) Comparison between quantum-Poincar´e sections [shown by color bar (grayscales)] and classical torus structure. The dashed lines display the borders of the classically permitted region. Since the Hamiltonian is a quadratic function of the momentum, there are two possibilities to choose P1 when calculating the section. Both solutions differ from each other in the sign of the velocity V1 in the U1 direction. Therefore, agreements with only one part of the torus structure can be achieved at any time. The color bar is chosen equal to compare both results. We set σ to the constant value σ = 2. All values are given in atomic units.
Finally, we want to estimate at which field strengths a solution of the classical structure may be possible. Since the classical orbit is localized in the z = 0 plane, we regard the extension of the torus structure around the orbit in a z − pz -Poincar´e section as its classical uncertainty (note that we change back to nonregularized coordinates in this place). Assuming that the quantum mechanically determined uncertainty product zpz = 1.59 for one resonance of the first series does not change to a great extent with the value of B, since it is already sufficiently close to the critical value
Following the quasi-Penning resonances up to high energies and field strengths, we could resolve their complete bifurcation behavior. We found out that only K1 , B2 , and three-dimensional orbits are involved in the bifurcations of B1 . It was shown that several phenomena appear in the region of F ≈ 0.5, E ≈ 0.6 including a cusp bifurcation between B1 and B2 as well as the appearance of a stability island for B2 . The reappearance of those phenomena when analyzing the stability and bifurcation behavior of D1 could then be interpreted as a possibility to find them on-and-off-again for orbits of even more complicated shape. The puzzle of the z-stability region of B1 ending in an apex in parameter space could be resolved by a closer examination of the stability of B2 . The results show a coincidence of two stability apexes at Sz,B , which indicates the continuation of the stability borders limiting the areas of stability in the z direction from one of those orbits to the other one. Semiclassical quantizations of B2 showed agreements with resonances found in exact quantum spectra. Due to further agreements between quantum-mechanical probability densities and classical orbits as well as between quantum-Poincar´e sections and classical torus structure, we have to conclude that signatures of the stable orbit B2 in exact quantum spectra have been found. Due to limited computer memory and power, it is not yet possible to reach regions of several millitesla in order to find out whether the widths of the detected resonances will disappear within the stability island or not. Nevertheless, we think that an experimental proof of B2 in photoabsorption spectra at the estimated field strengths will eventually be possible. Finally, since the widths of the observed resonances do not vanish within the stability island of B2 , and since the stability island is very close to the bifurcation lines with B1a and B1b , we think that the orbits B2 cannot be treated as isolated ones in a semiclassical quantization. It would therefore be preferable to perform a uniform semiclassical quantization [74–76] of the observed cusp bifurcation.
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ACKNOWLEDGMENTS
We thank Eugen Fl¨othmann for stimulating discussions.
CLASSICAL DYNAMICS AND LOCALIZATION OF . . .
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